Reaction rates for the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction

The magnesium-potassium anticorrelation observed in globular cluster NGC2419 can be explained by nuclear burning of hydrogen in hot environments. The exact site of this nuclear burning is, as yet, unknown. In order to constrain the sites responsible for this anticorrelation, the nuclear reactions involved must be well understood. The ${}^{39}\mathrm{K}+p$ reactions are one such pair of reactions. Here, we report a new evaluation of the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction rate by taking into account ambiguities and measurement uncertainties in the nuclear data. The uncertainty in the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction rate is larger than previously assumed, and its influence on nucleosynthesis models is demonstrated. We find the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction cross section should be the focus of future experimental study to help constrain models aimed at explaining the magnesium-potassium anticorrelation in globular clusters.

Figure 1

Comparison to experimental resonance strengths compared to the data of Ref. [32]. Uncertainties in the points include the uncertainties reported in Ref. [32]. Clearly, although there is agreement between resonance strengths at the normalization resonance at $E_r^{\text{c.m.}}=1992\mathrm{keV}$ (shown as a vertical dashed line), the agreement is poor at most other energies. There appears to be an overall systematic shift to higher resonance strengths in Ref. [32].

Figure 2

Comparison to experimental resonance strengths compared to the data of Ref. [32]. Uncertainties in the points include the uncertainties reported in Ref. [32] and additional uncertainty introduced by our renormalization process. In this case, the Kikstra resonance strengths have been reduced by a factor of 1.28 as described in the text. The new normalization points are shown by three vertical dashed lines. Although there is still a systematic disagreement between data sets, it is now within the $2\sigma$ uncertainties.

Figure 3

Example probability density functions for two experimentally measured resonances (at $E_r^{\text{c.m.}}=763\mathrm{keV}$ and $E_r^{\text{c.m.}}=1537\mathrm{keV}$) as and reported in Ref. [30] (“LEE66”), Ref. [31] (“CHE80”), and Ref. [32] (“KIK90”), respectively. By summing the probability density functions incoherently, the HDPI Bayesian method can be utilized to summarize our confidence in the experimental results. The dark and light shaded regions correspond to the 68% and 95% confidence intervals. The resulting log-normal probability density approximation is shown in the lower panels as a black, dashed line. See text for more detail.

Figure 4

Reaction rate probability densities for the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction. The reaction rate has been normalized to the median, recommended rate. Hence the thick and thin lines correspond to the $1\sigma$ and $2\sigma$ uncertainty bands. The color scale highlights that the rate probability distribution at each temperature is continuous with no absolute upper or lower limit. The solid green (grey) line represents the most recent calculation of Ref. [31].

Figure 5

Fractional resonance contributions to the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ reaction rate as a function of energy. A value of unity indicates that a particular resonance is responsible for 100% of the reaction rate at that temperature. The finite width of resonance contributions reflects their uncertainty calculated with the Monte Carlo method. To improve clarity, upper limit resonance contributions are shown by dashed lines at their 84% “high” contribution value. The solid line at high temperatures represents the summed contribution of all resonances that, individually, contribute less than 20% to the total reaction rate.

Figure 6

Monte Carlo nucleosynthesis results at $T=170\mathrm{MK}$ and $\rho =100\mathrm{g}/{\mathrm{cm}}^3$. The upper row represents results for the STARLIB reaction rates, while the lower row contains those obtained here by fully evaluating all experimental information for the ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ and ${}^{39}\mathrm{K}(\mathrm{p},\alpha ){}^{36}\mathrm{Ar}$ reaction. The increased correlation and importance of ${}^{39}\mathrm{K}(p,\gamma ){}^{40}\mathrm{Ca}$ on potassium production is seen in the two right-most plots.

Figure 7

Final abundances for potassium obtained for the Monte Carlo nucleosynthesis study discussed in the text. A KDE was found for the STARLIB rates and our reevaluated rates. The REACLIB rate does not define a probability distribution, so its predicated potassium abundance is shown by the solid black line.